2_Stresses in Shells

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Originally published by Springer-Verlag OHG_, BerliniGottingenIHeidelberg 1960
Softcover reprint of the hardcover 1st edition 1960

PREFACE
There are many ways to write a book on shells. The author might,
for example, devote his attention exclusively to a special type, such
as shell roofs or pressure vessels, and consider all the
minor details
of stress calculations and even the design. On the other
hand, he
might stress the mathematical side of the subject to such
an extent
that he virtually writes a book on differential equations under the
guise of the mechanical subject. The present hook has been kept
away from these extremes. At first sight i t may look
to many people
like a mathematics book, but it is that the serious reader will
soon see that it has been written by an engineer and for engineers.
In a theoretical subject such as this one, i t is, of course, not possible
to get very far with the multiplication table and elementary trigonom
etry alone. The ma,thematical prerequisites vary widely in different
parts of the book, depending on the subject. In some parts ordinary
equations with constant are
uct solutions of partial djfferential equations, the theory of complex
variables, or numerical analysis ",ill be en countered. However, the
author wishes to assure his readers that nowhere in this
book has
an
advanced mathematical tool been used just for the sake of displaying
it. No matter which mathematical tool
has
The book may be divided into
four parts. Chapter 1 contains
preliminary matter, and a reader sufficient,ly familiar with the basic
definitions may omit. this chapter until he finds that a real need for
studying it arises.
Chapters 2 to 4 contain the membraue theory, i. e. the theory of
shells whose bending rigidity may be neglected. The spectacular
simplification thus obtained makes it possible examine a wide
variety
problems of tanks
many examples of these applications have been included. There is,
of course, a heavy
the

chapters. I t has been considered important to show that
the
first solvirig the bending problem-a task
which often enough is
out
of reach of the practical engineer and even of the research worker.
Chapters 5
the two
important types of shells, the circular cylinder and the general
shell of revolution.
the development
rather elaboratc preparation,
a careful choice of subject matter had to be made; otherwise the
proper balance between simple complicated would have
been lost. In these two chapters an attempt has been made to
cover
and to carry every theory to
a definite
end, viz.
coordi
nates. In many cases it has been possible to present these results in
the form of a table. I t has, however, mostly been left to the reader
to adapt
a solution t,o his particular case of boundary conditions.
Chapter 7 is concerned with the stability of shells. From a research
man's point of view this is a rather unrewarding subject. A long
struggle through the
a tedious
numerical evaluation
ultimately yield a curve or only
a single numerical factor in a simple formula. And, after all, there is
only a rather 100Re correlation between the actual collapse of a shell
and the buckling
cases a large-deformation analysis has thrown
light into a dark corner
of our
book a choice of stability problems has
been
the author has been giving
for
many
much
of
what
is
found i this book goes beyond the possibilities even of an elaborate
university course.
Among these,
namely, design engineers and stress analysts who need shell theory
for their work, and research workers
entering the
field or working
in it. For the first group, the book offers a body of well-established
knowledge that will help
them in most cases
The second
and as a
l'R.KlfAVJ<.; v
for their own endeavors. The l m o t a t e ~ bibliography should be partic
ularly helpful
The author owes many thanks to colleagues and former students.
Professor S. TIMOSHENKO gave the
first encouragement to undertake
the big task, and he has followed its progress through the years with
steady
substantial parts of
Chapters 1 and 2 at a very early stage and gave the author much
good advice during his first steps of writing a book in English.
After
read it
many helpful suggestions. Parts of the manuscript have
been read at different stages by Drs. R. E. PAULSEN, F. T. GEYLING,
R. H. STIVERS, H. V. HAllNE, D.
A. CONRAD, P. M. RIrLOG,
F. A. LECKIE, and R. A.
EISENTRAUT, and smaller parts
by many
more of the author's students. All of them deserve the author's sincere
thanks for constructive suggestions and for checking formulas.
Los Altos, Calif., February 1960
w. Flugge
IN SHELLS . . . . . . . . 1
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Stress Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Membrane Forces.. . . . . . .. . .. . . . . . . .. .. . .. . . . . .. .. . . . . . . . . 8
D i r e c t i o n ~ . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Rectangular Coordinates... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Transformation of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter 2
2.1.1 Geometrical Relations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2.1 Spherical Dome. . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . .. . . . 24
2.3.1 Drop-shaped Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.1 General Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.2 Spherical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.2.1 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.2.2 Distributed Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.2.3 Edge Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.4.1 Solution by an Auxiliary Variable. . . . . . . . . . . . . . . . . . . . . . . 70
2.4.4.2 Solution by Numerical Integrat ion. . . . . . . . . . . . . . . . . . . . . . 73
2.4.5 Shell Formed
Shel l . . . . . . . . . . . . . . . . . . .
the
2.5.3.2 Axially Symmetric Deformation
3.1.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.1.1.1 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.1.1.3 Homogeneous Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U4
3.1.3 Barrel-Vaults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.2 DefoJ'Jllations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.2.2.3 Fourier Series Solutions for
the
3.4.2 Regular Dome
3.5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Chapter 4
4.1 Conditions of Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.2 Elliptic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.2.2 Elliptic Paraboloid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.3 Hyperbolic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.3.2 Hyperbolic Paraboloid, Edges Bisecting the Directions of the Gen-
erators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.4.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.4.2.3 The General Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.4.2.4 Polygonal Domes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4.4.2.5 Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
. .
5.1.3 Differential Equations for
5.3 Loads Applied to the Edges
x
5.4.2 Barrel Vaults
. . . . . . . . . . . . . . . 247
5.4.3.1 Isolated Boundary
. . 271
5.5.2.1 Homogeneous Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
5.5.3 Shell of Variable Thickness
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
5.6.2 Differential Equations for the Shell with Ribs.. . • . . . . . . . . . . . 307
5.7 Folded Structures 307
6.1.2 Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
6.2.1 Spherical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
6.2.1.1 Differential E q u a t i o n ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
6.2.1.2 Solution Using Hypergeometric Series . . . . . . . . . . . . . . . . . . . 324
6.2.1.3 Asymptotic Solution for Thin-walled Shells . . . . . . . . . . . . . . 330
6.2.1.4 Simplified Asymptotic Solution . . . . . . . . . . . . . . . . . . . . . . . . . 336
6.2.1.5 Bending Stresses in the y'icinity of the Apex . . . . . . . . . . . . 345
6.2.1.6 Concentrated Load at the
Apex . . . . . . . . . . . . . . . . . . . . . . . . 350
6.2.2.1 Elastic Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
6.2.3 Conical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
6.2.3.2 Example: Sludge Digestion
6.3.1 Spherical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
6.3.1.3 Oscillatory Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
6.3.2 Conical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
7.2.1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
7.2.1.2 Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
. . . . . . . . . . . . . . . . . . . . . 423
7.2.3 Solution for Shells with Shear Load
. . . . . . . . . . . . . . . . . . . . . . . 436
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
7.2.3.2 Shear and Axial Compression in a Cylinder of Finite Length 439
7.2.4 Nonuniform Axial Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
7.2.5 The Beam-Column Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
7.2.5.1 The Axisymmetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
7.2.5.2 Imperfections of Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
7.2.6 Nonlinear Theory of Shell Buckling . . . . . . . . . . . . . . . . . . . . . . . . 466
7.3 Spherical Shell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
478
BIBLIOGRAPHY
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
Table 2. Cylinder Loaded along a Generator . . . . . . . . . . . . . . . . . . . . . . . . . 244
Table 3. Barrel Vault
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Table 6. Functions
~ 2 " ••••••••••••••••••••••••••••••••••• 390

1.1.1 Definition of a Shell
Every part of a structure, of a machine or of any other object is a
three-dimensional body, however small its dimensions may be. Never
theless, the three-dimensional
of elasticity is not often applied
when stresses in such a body are calculated. There is a simple reason
for this: Every
of the most
point
to another. Cables, shafts and columns are typical examples of such
elements
~ h i c h
receive a force or a couple at one end and transmit it
to the other, whereas beams and arches usually transmit loads
to
sup
ports at both ends. The stress analyst does not envisage these elements
as three-dimensional
but rather as lines having some thickness, a kind
of "physical lines" as opposed to the mathematical meaning of the
word. When he wants
of
the
stresses acting
in it. Instead of describing this resultant by its magnitude, direction
and location in space, he usually gives its three components and
its
moments
about
normal force, two (transverse) shearing forces, two bending moments,
and
the
resultants" in
the cross section.
Not all structural elements are of the kind just described. A second
large group consists of all those which are made to bound or enclose
some space: walls, in the widest sense of the word, e. g., the wall of a
tank,
hull of an airplane, or the cloth-and-rubber hull of a
balloon. All these objects cannot be described by a line, but can be
described by a plane or curved surface, and consequently, their stress
analysis
must
be
built
on the concept of a "physical surface", a surface
made of some more or less solid material, capable of transmitting loads
from one part
In the development of the mathematical theory of such structural
elements, it has become necessary to distinguish between two types:
Flftgge. Stresses in Shells 1

Plane walls are called plates, while all walls shaped
to curved surfaces
called shells.
Summarizing these comiderations, we may define a shell as an ob
ject which, for the purpose of stress analysis, may be c o n ~ i d e r e d as the
materialization of a curved surface. This definition implies that
the
thickness of a shell is small compared with its other dimensions, but
it does not require that the smallness be extreme. I t also does not re
quire that the shell be made of elastic material. The occurrence of
plastic flow in a steel shell would not prevent its being a shell; a soap
bubble is also a shell,
although made
liquid, because of the surface tension acting in
it; has all the properties
of a true shell and
may be treated
and generally in
this book we shall assume that this material is elastic according
to
HOOKE'S
law.
In most cases, a shell is bounded by two curved surfaces, the faces.
The thickness t of the shell may be the same everywhere, or it may
vary from point to point. We define the middle surface of such a shell
as the surface which passes midway between the
two faces. I f we
know
the shape of the middle surface and the thickness of the shell for every
one of its points, then the shell is geometrically fully described. Mechani
cally, the middle surface and the thickness represent the shell in the
same way as a bar is represented by its axis and the cross section.
However,
not
every shell fits this description. A parachute, for in
stance, is made of cloth, i. e., of threads crossing each other and leav
ing holes in between. Nevertheless, it is a shell, and
the "middle sur
face" which represents it is fairly. well defined, although not by the
definition just given. However, the thickness t is not easily defined in
such a case. Another example of this
kind is culvert pipe used in high
way work. For most purposes it may be treated as a shell in
the shape
of a circular cylinder, and its middle surface may easily be defined.
The real pipe, however, is corrugated, and
in alternate regions all of
the material lies either on one side of the "middle surface" or on the
other. For some special purposes one may, of course, consider the corru
gated surface which really bisects the thickness, as the middle surface
of this pipe, but in many cases this is not done, and shell theory may
still be applied.
1.1.2 Stress Resultants
Before we can define stresses in a shell. we need a coordinate system.
Since the middle surface extends in two dimensions, we need two co

3
some system of coordinates x, y has been defined on the middle surface
so that
the lines x = const. meet the lines y = c.mst. at right angles
(GAussian coordinates). We may then cut an clement from the shell by
cutting along two pairs of adjacent coordinate lines as shown
in
the four sides of the element are normal to
the middle surface of the shell.
Since it is not always possible for the distance ds
x
or dsy between
two adjacent coordinate lines to be the same everywhere, opposite sides
of the element will differ slightly
in
purpose this difference is of no importance.
The front side of the element is part of a cross section x = const
through the shell and has the area ds
y
<. /
I
JC
Fig. 1. Stress resultants and loads acting on a shell element
area have a certain
ds
y
approaches zero, the resultant decreases proportionately,
and the quotient "force divided by length of section" has a finite limit.
I t is therefore reasonable to call this quotient the "stress "resultant".
I t
and may
be measured in lb/ft
or kg/m, for example.
For all analytical work we must resolve the stress resultant into
components. We choose as a reference frame the
tangent
,
another tangent to the middle surface at right angles to dS
l l
force components in these directions we give the following definitions:
In a section x = const, the force i direction x, transmitted by a
unit length of section (measured on the middle surface) is called the
normal force Nx '
I t is considered positive if tensile and negative if
compressive. The normal force
correspondingly.
1*
4 CHAP. I: STRESS SYSTEMS IN SHELLS
In a section x = const., the force transmitted by a unit length of
this section and directed tangent to dS
II
It
the direction of increasing y on
the same side of the shell element where a tensile force N z would point
in the direction of increasing
x.
Correspondingly,
l lz
is defined
itsposi
tive sign (fig. I). Evidently the sign of both shearing forces depends on
the choice of the coordinates. I t changes when the positive direction of
one of them is reversed.
In
to the middle surface
Qz.
The
the
result
ant
of a certain kind of stresses (fig. 2), normal stresses (O'z, 0'11)' shear
stresses parallel to
ll z
Consequently they also deserve the name of
"stress resultant", and so they will be called in this book.
The foregoing definitions apply
every shell, including shells in
which the faces and thickness are not defined. In the common case of
a shell consisting of solid
material
included between its faces, it is
possible to express the stress resultants as integrals of the stresses
acting on a section.
which are derived from the foregoing definitions, as the definitions
themselves. We shall now derive these integrals.
In
section is by definition N z ds
lI
which act on
ll
is of differential magnitude,
we may disregard a possible variation in this direction, but we have to
consider a variability of all stresses across the thickness of the shell.
I t
an
has
the
not simply ds
it is
II
+1/2
Nzds
ll
= JO'zds
ll
sides is dropped, this is the equation which

shearing stresses T",y and T",. must be integrated to obtain
the forces N",y
-i/2
(la-c)
ry
- / 2
The minus sign which has been added to the equation for Q"" stipulates
that a positive transverse force shall have the direction shown in
fig. 1, which is opposite to the direction of 7:",z in fig. 2.
1 .
We may
the same reasoning to a section y = const and write
three more equations for the other three stress resultants; we must, of
course, keep

of curvature, say Tx'
of
shearing forces. The difference between
N x v and N vz
vanishes only if
TV (e. g., for a sphere), or if Txv does
not depend on z. In a thin
shell
t and Z are small compared with the radii Tx '
Tv; then
the difference
may often be neglected.
center
we
must consider them. The moment of the stresses ax in a section
x = const. is referred to a tangent to the line element dBv of the middle
surface. The moment is of differential magnitude and proportional
to ds
by
M x is finite and
represents a moment per unit length of section. Consequently, it may
be measured in such units as ft.lb/ft
or in.lb/ft or
the bending moment of
thickness t, their resultant
section
the section
v
One may easily read from fig. 2 the relations
+l/2 +t/2
xv=- TXV-r-y-z Z
be considered as the definitions of the bending and twisting
moments. The minus signs are arbitrary and fix
the
sign convention
used in this book. (See also figs. V-I band VI-I b).
When the same ideas are applied to a section y = const., another
bending moment and another
twisting moment are obtained:
7
Again, as in the case of the shearing forces, the shear stresses in eqs.
(lh, j) are equal, but the
resultant moments are different. And again
the
difference is not large and may often be neglected (see p. 216), but
may sometimes be the
exact formulation of a problem
(see p.421). I t will be noticed that, because of the factors (rx + z)/r
x
and
are
inde
pendent of z, i. e., uniformly distributed across the thickness. TheEe
factors are required because of the curvature of the shell and represent
the fact that the sides of an element
are
on the middle surface.
that eqs. (lg-j) do not imply any
particular law
or not the
valid
xz
The
sides of a rectangular
the
shall call them
the
Chapters
2-6
of this book to explain the methods which ailow their computation
shells of different shapes.
known, the stresses may be found by
elementary
methods.
stress
obtain
the
stresses from the simple relations derived for beams of rectangular
cross section, subjected to a normal force and a
bending moment:
the
M
-term
is called the bending stress. I f the shell thickness is not very small
compared
with
the radii of curvature, it may be worth while to take
the trapezoidal shape of the cross section into account; but then one
should also make use of the basic ideas of
bars
pattern
(2c, d)

8 CHAP. I: STRESS SYSTEMS IN SHELLS
the discrepancy is not large in thin shells, we may usually disregard it
a t this stage of the stress analysis.
I f the bending and
twisting stresses are distributed linearly, the
transverse shearing stresses will have the parabolic distribution of the
shearing stresses
7:
zz
= 3 2 ~ Y (1 4 ~ 2 ) . (2e, £)
I f the shell ist not made of homogeneous material, or i f there is a
system of ribs or stiffeners incorporated into the shell, other formulas
must be In
problems arise.
1.1.3 Membrane Forces
Let us consider two examples of shells which behave very differ
ently.
First,
cylindrical form and paste the
edges together. This is a cylindrical shell. Very feeble lateral forces
will suffice to produce in it a considerable deformation. The resistance
of this shell
more complicated cases of this kind
the whole group of bending and
twisting moments may come into action.
Second, take the shell of an egg or an electric light
bulb. Both
they
visible deformation. In these shells a
quite
I t
forces N
would
expect the bending and twisting moments to be small, at least
in thin shells. A detailed study shows that
this is true.
While the first kind of shell is not very attractive for design pur
poses, the second one is, and whenever it is possible, engineers attempt
to shape and to support a shell so that it carries its load essentially
by
to
obtained in this way
is called the Membrane
the following Chapters, and we
shall see its merits and
its limitations.
I f the bending and twisting moments are zero, only the forces shown
in fig. 1 act on the sides of the shell element. In addition there may be
a load, proportional to the area
ds
z
• Mil
its
centroid
in an arbitrary direction. We shall now consider the moment
equilibrium of this force system. First, we choose as a reference axis

9
are those of the shearing forces N Zy and N yz'
The two forces N",
turning
counterclockwise if we look
"
N",y
ds
y
N",y = Ny",. (3)
Next, we choose the line marked "py" as a reference axis. I t is a tangent
to a line x = const. on the middle surface. With respect to this axis,
there is
y
intersect it, or they pass it so closely that
their moments are infinitely
moment
equilibrium for the axis "Px" we find in the same
way that
Qy = O.
Thus we arrive at a remarkable simplification of shell theory: Of
the ten unknown stress resultants, only three are left, N"" Ny,and
N", y = Ny",. The three equations of force equilibrium, which have not
yet been used, are available and sufficient
in
number for calculating
these forces (see pp. 19, 109, 167). When the normal and shearing forces
have
been found, the corresponding deformations may be calculated, and
we may check whether or not they lead to bending stresses. In many
cases it is found that
the bending stresses are negligibly small, and this
justifies the basic assumption of the membrane theory. In other cases
it is found that the deformations derived from the membrane theory
contain a discrepancy or a contradiction, and that, therefore, bending
and twisting moments must be an important part of the stress system.
When
brane forces (i. e., N
x
imply
that the normal
forces are necessarily tensile forces. In many shells they are compressive;
nevertheless the
theory.
1.2.1 Rectangular Coordinates
The membrane forces at a point of a shell represent a plane stress
system
the middle surface. When these stresses
or the stress resultants N x ,
Ny, N x y have been calculated for the sec
tions x =
const and
const passing through that point, the ques

arbitrary angle IX with the x direc
tion .
y
may
be found in textbooks on elementary strength of materials. We
need only repeat the
resultants of shells.
We consider a certain point of the shell (i. e., of its middle surface)
and define there two rectangular reference frames x, y and ~ , 1 ] (fig. 3a).
The directions x and y may be those of the GAussian coordinates used
on
the preceding pages for defining the normal and shearing forces N x ,
Ny, N
, and we assume now that these forces are known. We
wish to find the forces in sections ~ = const and
1] = const as defined
Fig. 3. Equilibrium of tr iangular shell elements
by the second reference frame (which need only be defined locally).
We obtain them
two sides ds
forces are known, and one side ds where two of
the
ing
equations:
x
the angle IX as shown in fig. 3a, we have
ds
x
s ~
and so we obtain the first and third of the following formulas:
N ~
= Nxcos
2
ex +
N
y
sin
2
x
2
(4a-c)
ARBITRARY D I R E C T I O ~ S
11
fig. 3
written
in
- Ny)sin2(X Nxycos2(X.
Eq. (5a) gives the normal force as a function of the direction of
the section. When IX varies through 180°, N ~
must have at least one
maximum and one minimum.
=
lXo for which
these extrema occur, from the condition d N ~ / d l X =
O. I t yields
are called the principal directions of the membrane
forces at this point
it
is zero for
IX = lXo' The extreme normal forces are called the principal
forces and are
maximum and
the other one the minimum that the normal force N or N I'} can assume
for any direction at this point. From eqs. (5) and (6) one may
obtain
Na = ~ (Nx + Ny) + ~ V(N
(7)
One of the principal forces makes an angle lXo with the x axis, the
other
one with the y axis, but eqs. (7) do not indicate which of them is N a
and which N
b
• To find this out, one must use either eqs. (4) or MOHR'S
circle (see p. 12).
When the principal directions are known at every point of the shell,
one may draw a net of curves which
have
indicate
the
to the supported edges by a
system of tensile and compressive forces in the shell. These trajectories
may give a very suggestive picture of the stresses in a shell (figs. II-26,
II-31), but they are laborious
to obtain
paper. Therefore they are not often used
in
in which direction a thin shell may best
be reinforced by ribs, and in which directions the steel rods in rein
forced concrete shells should preferably be placed.

1.2.2 l\'1ohr's Circle
just
i YXy), and the
moments and products of inertia of a cross section (1x ' 1y, 1x y). In all
these cases there exists a set of formulas identical with eqs.
(6)
and
(7)
service
as
LAND
's
MOHR 'S
circle appears
methods
normal and she aring forces
much of their former importance, we shall describe it here in some detail
because of its usefulness for the qualitative understanding of stress
patterns.
We
(4) for various sections passing
through this point. In a rectangular coordinate system we mark the
points x
Ny,
N x , respectively,
and then we draw a circle which has the line x y as a diameter (fig. 4).
The center of this circle has the coordinates
t (N + Ny), 0, and
its
radius
is
haye the
being zero. Consequently, the points
, y, a, b r;epresent the forces transmitted through sections which pass

Since the circle is unequivocally determined by the principal forces
N
a
, N
b
, we should necessarily have found this same circle, i f we had
started
from
the
pair of orthog
onal sections passing through the same point of the shell. Hence, this
circle is the locus for all points whose coordinates are
the
is a graphical
representation of the stress resultants at the particular point of the
shell. I t is called MOHR'S circle.
From
eq.
(6)
we see
that <9: x 0 a = 2(Xo, and from a well known
theorem of elementary geometry it follows that <9: x b a = (Xo'
In the lower right-hand corner of fig. 4 are shown the reference
frames x, y and
in
which
transmitted.
We may define a pole p on MOHR'S circle
by
drawing
through
one
straight
line parallel to the corresponding line
of the reference frame. All such lines lead to the same point p, and the
angle (xo is found again there.
When
~
draw
parallel to it the line p ~ through the pole p, we may
read from the figure the following relations for the coordinates of the
point ~ : Its
abscissa is
=
2tX,
i. e., exactly the normal force N ~ as given by eq. (5). The ordinate of
the point
is
=
fj •
Evidently, every point of MOHR's circle r.orresponds to one possible
section through
the
MOHR
that MOHR'S circle requires a sign convention
of
its
plotted to the right

downward when it was associated with N", and upward
when associated with Ny . We may easily verify the rule that
the
right
angle between the normal and shearing forces in a section and the
right angle between the directions in which they are plotted must
always be of opposite sense, one of clockwise and one
counterclockwise . As an example, we may look at
the forces N", and N",y
in fig. 4. At the shell element they point right
and up, in
On the curved middle surface of a shell the coordinates
cannot
be
simple cartesian coordinates but must be some kind of orthogonal
curvilinear coordinates. In many cases it is advisable to use, instead,
N d -- Ryds.
Y S,Jt I
to the general shape
of the middle surface or to the boundaries of the shell (see Chapter IV).
In such cases the lines x = const and y = const meet each other at
an angle w which may be constant or even vary from point to point.
The shell element is then in the first approximation a parallelogram
(fig. 5).
The membrane
in
middle surface. There
nents. One
might think of using rectangular components N " dsy and
N "" y day. These correspond to the definitions of normal and shearing
forces given on p. 4, if we use a rectangular reference frame x',
y.
The
adjacent side of the element should then be resolved
into the rectangular components Ny' ds", and Ny'
" ds", shown in fig. 5,

15
The two shearing forces Nx'y and Ny'x are, of course, not equal since
equality can be expected only for sections at right angles to each other.
Therefore, the tensor of the membrane forces is now described by four
quantities insted of three. These four quantities, however, are not
_--ds.
y
moment equilibrium with respect to a normal to the shell:
N
x
' ds
y
' ds
x
cosw
- Nx'Yds
y
' ds
x
(8)
We may avoid complications and arrive at a more natural descrip
tion of the state of stress at a point (i. e., of the membrane force tensor)
if
of the lines x = const and y = const (fig. 6).
On the sides ds
per unit length the "skew fiber force" N x
and
the
"skew
shear Nx'y but
tween the orthogonal and the skew forces:
N _
N
x
R
y
and N
N ", or Ny
on the
same line
and do not yield a couple. Thus the shearing forces are again alone in
the equation of moment equilibrium:
and hence
N", y = Ny ", .
Having solved a shell problem in oblique coordinates x, y, we may
desire to find from the skew forces N"" Ny, N", y the component,s N ,
N'I' N 'I for an orthogonal
pair
N
b
• The set of transformation formulas needed may be found by
the
'1\ "./
y
~ x
w
the
other two sides parallel to the directions x and y (fig. 8). The equi
librium of all forces in the direction ~ vields the equation
N ~ d s ' l = N " , d s y c o s O i . ~ + N",ydsy sinOi.'1 + Nyds", sinOi.'1 + N",yds", COSOi.. ,
and a similar equation will be found for the 17-components:
+
Between the three sides of
the element we have the
geometric
relation
sinw
~ = ~ = ~ .
the three following, equations,
element:
2
01.7] + N x y c o s O l . ~ s i n O l . 7 ] '
N7] sinw = N
2
s i n O l . ~ COSOl.7]'
N ~ ' 1 s i n w = NycosOl.7]sinOl.7]- N x c o s O l . ~ s i n O l . ~ +
(9)
+ x y ( c O S O l . ~ C O S O l . 7 ] - s i n O l . ~ s i n O l . 7 ] ) '
To find the principal forces N
a
This is an equation for the unknown angles l X ~
and
bring this
n
formation leads to an equation in which only
the functions COS2lXa
t 2 Ny
t 2 Nx sin2ro +
(lOb)
Iectangular coordinates.
asked and answered on the preceding
pages for the normal and shearing forces may also be formulated for
the bending and twisting moments. The answers may be
found
easily
by reducing each moment problem to the corresponding force problem.
We simply replace each
moment by a couple of forces parallel to the
middle surface. The arms of all these couples must be equal, but are
otherwise arbitrary. We choose them equal to the thickness of the
shell. We
have then in its upper surface a system of normal and shearing
forces and in the lower surface a system which is
identical except
that
the direction of each force is reversed. We may now cut triangular and
other elements from the shell and write fwo each one of the two force
systems the equations
the preceding sec-

tions. The resultant forces
bending
and
twisting
moments.
tion 1.2 we
valid results for the transformation of the bending and
twisting
moments
to
2.1.1 Geometrical Relations
The particular type of shell which we are going to treat in
this
struction of tanks, pressure vessels and domes.
Before we enter into
in
_ _ __
by the
axis in
shell of r'evoiu tiO Il
its plane. This
by
it is found and by
giving the value of a second
coordinate which ' varies along
around
to each other, they are called the "parallel
circles"
lar distance eof its plane from that of a da
tum
meridian
and
choose
as
its' axis of revolution. I f the middle surface
of
our shell is a sphere, these coordinates are the spherical coordinates
used
in
is the complement to the
latitude: hence the nomenclature of the meridians and
the parallel
2.1 GENERAL DIFFERENTIAL EQUATIONS 19
Fig. 1 shows a meridian of the shell. Let r be the distance of one
of its points from the axis of rotation and r
1
2
, measured on a normal to
the meridian between its intersection with the axis of rotation and the
middle surface. I t is the second radius of curvature of the shell, and
we read from fig.
have
we have the relations
1 dr rl
(5)
The shell element (fig. 2) is cut out by two meridians and two
parallel circles, each
indefinitely close together. The conditions
of its equilibrium will furnish three equations, just enough to determine
the three unknown stress
shear N</>o.
tangent to the meridian.
The shear transmitted by
one of the meridiona1
edges of the elemcnt
opposite direction and
therefore almost cancel each other. Only
their
difference
8Na+
1 dO d<f>
enters the equilibrium condition. In the same way we have the diffe
rence of
bear
in mind that both the force per unit length of section, N +, and the
length of section r dO vary with cf>. Therefore we have to introduce the
increment
8
into the condition of equilibrium.
But that
l
dcf>
on either side of the element
lie in the plane of a parallel circle where they include an angle dO.
They
the
shell. We resolve this
force into two rectangular components normal to the shell and in the
direction of
N ar
d<f> dO •cos cf>,
enters our condition of equilibrium, and since its direction is opposite
to that
N+,
Finally we have to
the external force, which
is the product of the load component per unit area of shell surface, p+,
and the area of
-
+
the
ing by this, we get the partial differential equation
8 8Na+
= O. (6a)
By quite similar reasoning we obtain an equation for the forces in
the direction of a parallel circle. For the difference
of the two shearing
forces which are transmitted in the horizontal edges of the shell element,
we must take
element:
Then we have a term representing the difference of the two forces
Na • r

21
we have a contribution from the shear acting on the meridional edges.
The two forces
N o ~ . r
components make an angle dO and therefore have a resultant
N o ~ ' '1 dep. cosep • dO
which has the direction of the tangent to a parallel circle and thus
enters our equation. I f we drop the factor dO dep, common
to
we
have:
8
8N
o
a ; j ; ( ' N ~ o ) + l---a8 + l N o ~ c o s e p + porr
1
The third equation refers to the forces which are perpendicular
to
I t
that
resultant
NIJ '1
dep d()
its
N ~
and the
Pr , '1 dO dep
of the load must be in equilibrium. This yields the equation
NIJ 'lsinep
of our equations:
(6c)
This equation not only is valid for shells in the form of a surface of
revolution, but may be applied
to
all shells when the coordinate lines
ep = const and 0 = const are the lines of curvature of the surface.
Therefore, we shall meet it again in the next chapter, and we shall see
in Chapter 4 what becomes of it when the coordinates no longer follow
the lines of curvature of the shell.
I t is notable that eq. (6c) does not contain any derivatives of the
unknowns. I t
the
and to reduce our problem t two differential equations,
with the shear and one of the normal forces as unknowns.
Till now, we have used two angular coordinates 0 and ep. This is

done
quite
generally in the theory of shells of revolution. However the
angle </> is very inconvenient if the meridian
has a point of inflection.
At such a point, </>
crease. The stress
resultants must therefore be double-valued functions
of </>, the two branches belonging to the two parts of the
meridian above
that
depends on the direction in which </> increases. Since
this is reversed beyond the inflection point, the shear must suddenly
have the opposite sign, without passing through zero. I t is evident
that an analytical solution fulfilling all
these
requirements
cannot
be
the
differ
ential
equations will also meet with difficulties. For such cases it is
useful to replace </>
these difficulties,
and that
from
any
datum
the
wise from its edge. Consequently, we then replace the subscript </>
by 8 .
Between 8 and </> we have the relation (2) and introducing this into
the
(7 a-c)
formulating the
plane of the meridian (fig. 1).
From (4b) we find that
a . a
and when we introduce that into (6a, b), we find
(8a, b)
There is
equations,
if the
shape of the meridian is given by its equation in rectangular coordi
nates r, z. However, there is no particular reason to prefer for
struc
2.2.1 Differential Equations
metry as the shell itself. Then the stresses
are
this coordinate disappear from eqs. (6).
The equations (6a, c) then
read
d
only
about
its
We eliminate it from
our further considerations by putting Po == 0 and Ncf>o - O. When we
solve eq. (9b) for
a
first
left
may
:e/> (rNcf>sincf» = d ~ (r
2
by
1
r
2
{Pr
be found from (9b).
Eq. (10) may be interpreted as a condition of equilibrium for the
part
of
the
cf> =
shell along this circle, 2n
r
2
sincf>
2
sin
2
cf>
in this section.
The integral times 2n represents the distributed load applied above this
circle, if we write it as a definite integral between appropriate limits.
The upper
cf>
for
the
cf>
flat top (figs. 6, 14) we
have
applied above the circle cf> =
cf>o (see fig. 4), 2n
0
24 CHAP. 2: SHELLS OF REVOLUTION
I f the shell is closed at the vertex, such an additional load can only
be a concentrated force P applied at this point. I f no other load is
present, we have in eq.
(10)
N =- P
(11 a)
(11 b)
At the
top both forces have a singularity of the second order, i . e.
they
tend toward infinity as </>-2. We shall see later (p.
350) that the
load is applied
but that at some distance
·the membrane forces as given by eqs. (11) still represent the real state
of stress.
2.2.2 Solution for
some Typical Cases
2.2.2.1 Spherical Dome
As a first example we consider a spherical dome as shown in fig. 3.
We ask for the stress resultants produced by a dead load p (weight per
unit area
its components tangential and normal to
the shell. These are
Introducing this into (10), we find with r
1
= T2 =
a:
2
cf> .
o
(12)
resultants the formulas
(13)
I t is interesting to discuss these forces in some detail. When we put
</> = 0, we find N = N e = - p
a/2. The meridional force N is nega
tive
throughout
with increasing
</> = 51.82
0
25
cf> + cos cf> - 1 = o.
I f the shell is so flat that cf> does not exceed this limit, no tensile stres'>
appears, assuming that the
dead load is the only load and that a proper
abutment is provided. This abutment has to resist the thrust N.p which
has the direction of the tangent to the meridian. Such an abutment
usually consists of a continuous vertical support and a ring, which
resists the horizontal component of N</> and from it receives a tensile
force
2
cf>.
This ring is the source of a pertUl bation of the membrane stresses given
by our formulas. In flat domes its stress is of opposite sign to the hoop
stress the shell, and high domes, where the hoop stress the
springing line is positive, it is usually much smaller than the stress in
the ring. Therefore, after
the elastic deformations, the ring and the
shell do not fit together. The continuity of deformation is re-established
by an additional bending of the
shell, which will be treated
in
here that the bending stresses
are confined to a border zone
of limited width and
fact, the simple stresses given
by the membrane theory.
4. Shell dome skylight
superstructure, the lantern. Its say2n· a sincf>o,
acts on the upper edge of the shell as a vertical line load. Since the
shell can resist only tangential forces, this edge also needs a stiffening
ring, which takes the other component (fig. 4) and gets a compressive
force from it. We find the stress resultants in such a shell with its
own dead load p and
the lantern load P
by returning to the integral (10)
and determining 0 so that for cf> = cf>iJ we have N.p = -Pfsincf>o. The
simple
computation
N = _ a cosCPo - coscp _ P sinCPo
.p p
The difference of the two cosines is disadvantageous for numerica

mulas in
CPo sin - = - c J > ~ __ p
sincpo
No = -N<b - pacoscf>.
Some figures may interpret this result. The roof represented in fig . 5
carries a uniformly distributed load p = 45lbjft
2
lbjft
, applied along its center line, i. e. on a circle
of 13' 5" radius. The edge of the shell
has
r = 13' 10", and the vertical line load P
transmitted at this edge is
correspondingly smaller,
<b - sin2cp
No =
At the upper edge (cf> = 9.96°) these formulas yield N4> =
-25801bjft,
No
38.7°): N<b =
upper
-107.5 Ib/in
500 or 600 Ibjin
stresses.
2.2.2.2 Boiler End
Pressure vessels of all kinds are built as shells of revolution, con
sisting of a cylindrical drum and two ends which may be shaped as
hemispheres, half ellipsoid or in
any other suitable form. They have to
resist an internal pressure p, constant and perpendicular to the wall.
When
we put P4> = 0 , Pr = p, the integral (10) may be simplified
considerably. Making use of (3a), we find
4>
N4>=
27
and
this integral may be evaluated independently of the shape of the
meridian. Eq. (9b) then yields the hoop force No. Thus we get
the
resultants in pressure
vessels:
(15)
We shall use these formulas to study some typical forms of boiler ends.
Boiler ends are often shaped as flat ellipsoids of revolution (fig. 6).
As we find easily by well-known methods of analytical geometry, the
---If ;
Fig. 6. Ellipsoid a s boiler end
elliptic meridian has the radius of curvature
a
2
b
2
curvature
pa
2
1
No. This is
ellipsoid but is true for any surface of revolution. At
the vertex all
meridians meet, and any direction is parallel to one of them and
at
right angles to another. Since in a surface of continuous curvature we
have at the vertex r
1
= r
2
tudinal forces
>or _ P, I ' l
~ - . H t J - 2 -
and this may be used as a boundary condition to determine
C
of the stress resultants in the shell.
The hoop force changes 'sign and becomes negative near the equator.
The zero is found where . . b
smcJ>
= V .
a
2
This formula yields a real angle only if alb 2
V2. I f the
ratio
of its axes,
an equatorial zone
exists where the hoop stress is a compression. The elastic deformation
of such a shell must be such that
the diameter of its border decreases.
On the other hand, the cylindrical part of the boiler has a posithe
hoop force
by putting
a. On the parallel circle where the two parts meet,
they have quite-different deformations and will
not fit together without
by bending
stresses, which
them in detail in
Chapters 5 and 6.
The discrepancy of the hoop forces of the boiler end and
the boiler
drum may
be avoided
0
of course, many curves which fulfill this condition. One of
them
may
(r
2
•
This curve is rather lengthy and therefore not particularly fit for
the
end of a pressure vessel, but its property of zero curvature at
z = 0
is preserved when we subject it to an affine transformation, sub.
stituting n z
a boiler
meridian,
we

[r2 (a
[r2 (a
+
29
Fig. 8 gives an example of the distribution of the stress resultants in
such a boiler end. I t shows the continuity of the hoop force. There is
I
I
l = =
Fig. 8. Boiler end without discontinuity in the hoop forces
N /J
a small zone in which No is negative. This may be avoided by choosing
n < 1.9. I f n is chosen much greater than 2, the compressive zone is
wider
and
2.2.2.3 Pointed Shells
I t is not necessary that the meridian meet the axis of the shell at
a right. angle. I f it does not, a shell with a pointed apex results. Such
shells have some particularities which we shall now study in a typical
example. The meridian of the dome, fig. 9, is a circle whose center does
not lie on the axis of revolution. Although the radius of curvature
/'1 = a of the meridian is a constant, the radius of transversal curvature
is variable:
= sin</> = a 1 - . sin</> .
We ask for the stress resultants produced by the weight of the struc.
ture, assuming a constant wall thickness. The load is then given by (12).
We find N from eq. (10) and a void the determination of the constant C
from a boundary condition by using the mechanical
interpretation
of
between the
sm - sm 0 sm .
(9 b) :
At the vertex
indefinite.
We
0
and
denominator
stress
distribution
is
shown in fig. 9.
In the limiting case </>0 = 0 the ogival dome becomes a sphere,
preceding formulas give stress resultants of a spherical
dome. In this limiting case N.p and No are no longer zero at
the top.
the limiting case is
when </>0 -+ 0:
For very small values of </>0' the normal forces rise rather
suddenly from zero
formulas, but it
to almost
of stress
by additional bending stresses, as will be discussed in Chap
ter 6.
the
meridian
begins
say </> = -
</>0 '
type of fig. 10, having a downward point
at
its

31
.p
o
sincf>o)
such a value that the
numerator vanishes too,
had
in
the ogival shell. Fig . 10. Shell requiring central su pport
But then N.p would be
infinite on the whole top circle cf> = 0, and
that would be much worse.
We choose tentatively C = 0 and we shall see
at
and
(9 b):
No = P2
cos2cf».
To study the singularity, we cut the shell in a parallel circle having
a negative cf>, say <f> =
-cf>' <
I t
is a
2 ( . A-,
.
is zero. This means that the meridional
forces there which are horizontal, cannot carry any load from the inner
part
of
the
therefore
find
that is possible only in the
center. Indeed , for <f>' = cf>o, the resultant R is
Ro
= pa
2
lesin
2
and
this
indicates
what
the singularity of the stress resultants means:
that forces of infinite intensity, acting on a circle of radius zero, carry
the total load applied on the
part
circle.
vertical
force Ro is needed
there. Then the infinity disappears if the thin shell extends only to
the circumference of this column.
The stress system, which we now have found, shows
nothing
special
on the top circle cf> = 0 and seems to be quite harmless. But on p. 99,
when discussing
deformations of toroidal shells, we shall see that
this stress system cannot be realized because it would lead to
an
im
bending
stresses
in
only to remedy an impossible deformation,
they
in
and which would
of the total
load. I t is this argument which finally justifies our choice for
the
rotation of a closed curve about an
axis passing outside. A toroidal shell encloses an annular volume and
may be considered as a pressure vessel. Figs.ll
and 12 show meridional
The shell, fig. 11,
as indicated by
the broken
line. The meridian of each part begins and ends with a horizontal
Figs. 11 and 12.
Toroidal shells
tangent. Therefore, the meridional forces acting at each edge do not
have a vertical com,ponent and
cannot transmit any vertical force from
one half of the shell to the other.
Now, when t.he shell is filled with gas
of pressure p, this pressure has a downward resultant on the
inner half
on the outer half,
2.2 LOADS HAVING AXIAL SYMMETRY 33
and neither part can be in equilibrium under the action of the pressure
P and the forces on its edges. I t follows that a membrane stress system
with finite values N 4>' N J is not possible
in
this
load.
This difficulty disappears when the two top circles have the same
radius, e. g. when the meridian of the shell is a circle (fig. 12). Then
eq . (10) gives with P4> = 0,
Pr
= p:
N4> = ( a s i n < l > ~ a R ) S i n < l > [f(aSin</>+R)Cos</>d</>+O]
= ( a S i n < l > ~ a R ) S i n < l > [- : (cos
2
</> - sin
2
</» + Rsin</> + oj,
and here we can determine 0 so that the singularities at 1> = 0 and
at
pa
water
sphere,
supported
AA.
support at 4>. =
120·
load for a water tank is the pressure of the water
(specific weight
y).
I t is normal to the shell (P4> = 0) and proportional to the depth. I f the
tank is completely filled,
CHAP. 2: SHELLS OF REVOLUTION
Bya simple integration, we find from eq. (10) the meridional force
Ncb = s ; n ~ :
I' a
2
c/>
+ 60).
To
obtain a finite value
of Ncb the factor in brackets must also become zero. This leads to
0 = 1/6, and after some simple transformation we find
I' a
above the supporting circle <P = <Po' In the
lower part of the shell we have to apply another value of 0, which
makes Ncb finite at <p = :n;. I t
is 0
ya
2
Ncb
=-6-
2
No
The distribution of these forces is shown in fig. 13.
The location of the supporting circle does
not
two
values of O. I f we give it a higher or lower position, only the domains
of validity of the two
pairs of formulas
changed. The corresponding
changes in the stress resultants are indicated by dotted lines in fig. 13.
=
in the meridian, which in a thin-walled structure
like this one should be avoided, and that a higher position cuts off the
peak
value
but
support.
discontinuously. The difference of the meridional forces is a load
applied to the ring. We resolve it into a vertical component
21' a
its numerous
into a horinzontal component
-3- sincf>o'
which is a radial load applied to the ring, producing in it a compressive
hoop stress.
35
Here we have again a case in which the direct stresses
lead to a
the continuity
of the struc
ture. A discontinuity in the hoop force means a discontinuity of the
elastic extension of the parallel circles. A membrane-stress system
which avoids
used
all available constants to fulfill other, more important conditions. The
continuity of deformations can be reestablished only by
an additional
bending of the border zones of both halves of the shell, and again we
have
A similar disturbance, but of greater
intensity,
is caused by the
connection of the shell to the supporting ring, if this is supported
by
vertical forces, as shown in fig. 13. Then the ring is subject to compres
sive stresses which fit the positive hoop stresses in
both parts of the
0
preferable to support the ring by inclined
bars, tangential to the meridians of the
shell, or even by a conical steel plate. Then
the ring is relieved of its hoop stress
causes less disturbance of the membrane
forces of the shell.
writing
---f
o
14. Spherical tank bottom
in
fig. 14. The evaluation of the integral (10) and subsequent appli
cation
at the edge of the shell there
must be a ring to take care of the horizontal component of the meridional
force N ~ .
in fig. 15a.
I t
is the lower half of an ellipsoid of revolution. Some for
mulas concerning its
add here the relation
The
load on the shell is Pr = y (h + z). When this is introduced into
the integral (10), a somewhat lengthy integration must be performed.
3*
I t remains, however, within the domain of elementary functions and
yields finally
2
sin2</> +
b
2
I t
of much use to do this
in general terms, since a rather clumsy formula would result. We. prefer
to write simply
At the bottom of the
tank, cJ> = 180°, we obtain
ya
2
At the edge, cJ> = 90 0, the meridional force is
Nq, = Y6
This force the whole weight to cylindrical wall.
We shall see on p. 195 how it may be transferred from there to a support.
The hoop force at cJ> = 90
0
is
I f b > aNi, this
may be positive when h is large enough, but it always
becomes negative when the water level in the
tank is lowered. I f
b < a/V 2" , the hoop force at the edge of the bottom is always a com
pression, independent of h.
= 1.5a, b
O.6a the stress resultants
are plotted in fig. 15b over the horizontal projection of the meridian.
-- ---1
Fig. 15. Ellipsoid as tank bottom, (a) Tank, (b) Tank
bottom
= No at the center and that the hoop
force changes sign
the edge of the shell. The greatest compressive

37
2.2.2.6
Conical
Shell
In conical shells, the slope angle cf> is a constant and can no longer
serve as a coordinate on the meridian. We replace
it by
use eqs. (7a-c). Simplifying them for axial symmetry andputtingcf> = ex,
r = 8 (OS (X, r = 0 0 , r2 = S cot ex, we find
from them
No
intensity the normal there
Fig. 16. Conical shell
no chance of adapting it to a boundary condition. A similar situation
exists
in
in
obtain
by
shelter
At
this must give zero, whence
C = P l2 /(2sin(X) which gives
_ p l2 _ 8
Fig. 17 illustrates
becomes infini te of the first
order, as we must expect at
a point support.
I t may easily be checked that the vertical resultant of the forces Ns
transmitted in a parallel circle approaches
the total load
of the shell,
s =
o.
I f the shell is not supported at the
center but
hoop force No will be the same, but in the
general expression for N.
force finite at s = O. This yields
N = - ~
8 2sin<x •
Of course, the support must be adequate to resist the thrust of the shell.
I f it can resist only
vertical forces, a pure
is not
possible in the shell, and the additional bending will be of such
magnitude
the border zone.
When we assume all distribute: loads to
be zero,
a concent.rated force applied at the top of the cone
in the
direct.ion of
as may
be found by integrating the vertical component of N. along
an
For the bottom
and
in
given on p.35. For the cone we
have with
N 8 = Y c;t X [I h2 - s sin X)
s ds + C] .
The constant follows from a condition
at the outer edge
tank and from
vertical
be
load P cotiX, producing the hoop force
F = P 1
is
N = y cota [2 (1
3
_ 8
3
bottom meet, a ring
must be provided which resists the difference of the horizontal compo
nents of the meridional forces N..
in the cone and N 4> in the sphere. This
ring
structural
pur
poses, if the dimensions of the shells are so chosen that the thrusts
of
cone and sphere balance each other. This condition can, of course, be
fulfilled only for a certain load, e. g. that one belonging
to
2.3.1 Drop-shaped Tank
In
the upper part of a spherical water tank, fig. 13, the meridional
stress G<b = N4>/t is rather small
compared with the hoop stress Go =
No/t.
t
all and
the hoop
stress alone carries the load. From eq. (6c) one might expect to save
some steel, if one could shape the tank so that
at every point the two
stresses
are
of constant wall thickness this amounts to having everywhere
N4>=N
tank is
to that
of finding the shape of a drop of liquid a plane surface.
N > =
No is then the capillary force which, exactly as the stress result
ants in the tank wall, must be
in equilibrium with the hydraulic
pressure. Therefore we speak of a d·rop-shaped tank.
To establish
meridian, we assume
that the tank is completely filled with a liquid of specific weight y and
that
at the highest point there is still a pressure, which we call
y h.
a safety valve or a standpipe, connected
anywhere with

top
nates r, z
In this notation the load acting on the shell
is
Pt/> = 0, Pr = Y z.
Since we want constant forces Nt/> = No = (J t, eq. (9b) becomes
( J t ( ~ + ~ ) = y z , (18)
l r2
while (9a)
Fig. 19. Drop-shaped tank
We now use eqs. (4a) and (1) to express the radii r
1
and
r
2
in
eq.
(18)
dsincb + sincb = "' "z.
as the
the solution of this
numerical
as useful as possible,
by putting
following form:
dTJ TJ
de YJ
41
and e
the derivatives
and e
step
by step to higher values of e. This method is rather crude, but in many
cases good enough. I t may be refined in many ways which are explained
in books on numerical integration
l
All these methods progress easily once the computation has been
started, but they require some outside help
in the beginning, the more
refined methods requiring more help.
We know that for e= 0 the tangent to the meridian must be hori
zontal,
that
z
introduce this into eqs. (20a, b), we find
that 'YJle assumes the form 010.
may that at the
surface of revolution r
20't 2a
r that
dl] h h h
For the more elaborate methods of numerical integration it is
to 'YJ
starting
the
following way.
Near the top of the shell sin</> is small, and may be neglected, com
pared with
1 in
in YJ
this
tion results:
E.:
ChapterV. New York 1952. - COLLATZ, L.: Numerische Behandlung von Diffe
rentialgleichungen. 2

Except for the minus sign this is BESSEL'S equation, and (21) is solved
by the modified
(23a)
and consequently
(23 b)
From these formulas 'Y} and Cmay be computed for small values of ( ,
and with them
may be started.
Whatever method may be used for the integration, it will be found
that it does
This is easy
understand since ( is not suitable as an independent
variable when cp approaches 90°. I t is therefore necessary to change
the procedure and to write everything in terms of C. From eq. (4b)
we derive that
I d coscp
r l = - ~
and from this and eq. (1) we see that eq. (18) may be
brought
into
the
form
d
pair
and z. As before, we use ( and
r; as dimensionless variables, but
instead
of
'Y)
~ = coscp.
d ~ VI- ~ 2 de ~
VI _ 2 •
(24a, b)
These equations may be used to continue the computation o cp ~ 140°,
and
the
to the set (20).
Fig. 19 shows a typical shape of meridian obtained by the proce
dure described. The meridian ends at
the point E with a horizontal
tangent,
cp
bottom into
<b
from the edge of the shell is introduced as a radial
load
and which, therefore, is in the same state of uniform biaxial
stress as the shell. However, the flat bottom cannot resist the vertical
pressure 'Y
say
43
a concrete "lab, which can exert on it an upward pressure of the same
magnitude. The total reaction of the support is the
product of this
pressure and of the area of the plane bottom, y H . 11: R2. Since the tank
is in equilibrium, this force must be
equal
y
Comparing this value with the volume obtained from direct integration
of the meridian
offers a valuable check for the numerical integration.
The computation depends on only one parameter hja. When it is
repeated for a series of different values of this parameter, a series of
Fig. 20. Series of drop-shaped tanks
~ ~
I
I
I
I
I
curves is obtained from which, by mere change of scale, all possible
solutions of the problem
series in which the scale for each curve
has
the
top
the greater the curvature of the meridian and the smaller
the tank. The small tanks are almost spherical, while the larger ones
are rather flat. This corresponds to
the fact that a small drop of water
on a plane plate looks
almost
a shallow puddle of almost constant depth.
For h =
Y) == C= 0 as far as
they are applic

cease to be so. The shell degenerates into a plane plate, and no tank of
reasonable shape can be obtained. Therefore, drop
tanks are not
built
th
hot weather.
specific weight
the
V. The first three
the parameter
has to
start the
computation with
vs . h/a
the volume of the resulting tank . Then the computation must be repeat
ed with a better fitting wall thickness until agreement between the
resulting and the required capacity is reached. This will be facilitated
by fig. 21, where Vlh
3
is
The drop-shaped tank cannot be expected to have uniform stress
if the actual load is different from the design load, whether it be that
the top pressure is not exactly as assumed or
that
the tank is only
partially filled, with or without ,some gas pressure on the liquid level.
In all these cases N ~
must be found from eq . (10) by numerical inte
gration and No from eq . (9 b). One essential result of such computations

45
work: When we separate the tank from its foundation, we find two
external forces acting on it, the weight of the contents and
the upward
force exerted by the foundation on the flat bottom. The latter force
equals the pressure 'j'
area of the bottom
arbitrary loading conditions one
cannot expect that this
reaction and the weight will be equal. The force N at the edge E,
fig. 19, cannot take care of the difference because it is horizontal, and
therefore a transverse shear Q ~ is needed at the edge. Since the mem
brane theory denies the existence of transverse shears, it will yield
N ~ = ±(X), and Ne will then become infinite, with' the opposite sign.
The practical application of the drop shape should therefore be limited
to the
or
Sp her ical roo f
~ - - ' - -
tangent
depends on the liquid to be stored. Tanks of great
capacity will always become rather flat may not be able to
support
their own weight when empty. Such tanks may be built in an open
form and closed by a roof which is not touched by the liquid (fig. 22).
The calculation of the shape is very similar to that described here.
The meridian starts with
a set of finite values e, 'YJ, C· I f YJ is small
enough, the approximate formula (22) may be used to start the inte
gration, but now both terms must be employed, and the constants
A, B
the initial values of 'YJ
and C.
A shell dome looks almost like a three-dimensional arch structure.
This raises the question whether or not for a given load there also
exists a best shape, analogous to the funicular curve for the arch.
This question shows plainly the fundamental difference
betw
een

seen
have equilibrium without bending in almost any shell for
almost any load, and the additional bending which occurs in boundary
zones is of somewhat the same importance as the bending moments in
a statically indeterminate funicular arch.
From this situation it
follows that we can
of bending. We can
mem
brane
point and every
direction.
As a first problem of this type we determine the shape of a dome
which has to carry
own weight. The problem is a simple one if the
dome consists of a plain concrete shell without additional
dead load.
specific weight of
of
the
and
putting
(here
a is considered positive when it is a compression, contrary to
our usual convention), we get
a t ( ~ . +
1
be transformed into
a simple differential equation for r(</» or, better, for </> (r):
de/> L _ t an t
dr r
may be solved by numerical integration, beginning at the vertex.
There we
have r
de/> 1 Y
d r = -;:-; = - 2 ~ ·
the
r
The wall thickness follows from eq. (9a), which here assumes
the
form
dt y dr
# = --u ttane/>. dq, .
This equation has a simple solution, when we transform it t rectangular
coordinates r,
dr
liZ =
cote/>
The solution is represented in fig. 23. The shell may be extended to
greater angles e/>, but then
the exponential growth of t leads to struc
tures which soon cease to
be thin-walled and prob
of technical
of
constant
cause shells of any reason-
able shape will have direct stresses far below the admissible limit.
But we see, for example, from eq. (13) for a spherical dome, that
the
therefore the stresses caused
by the weight of the shell are proportional to y
a.
This indicates that
they increase in proportion to the diameter of the dome, independ
ently of the wall thickness. Therefore, for every shape of the shell
there exists a certain size beyond which it can no longer be built in a
material of a given aly, and the shell of consta,nt stress is that which
allows the biggest dome.
Usually a large dome will have an opening at the top. We may use
the previous solution also in this case, i f we choose the thickness
t
so
that N</> = -a t, together with a compression ring, will be capable of
carrying the loads applied at the upper edge. But this is not the most
general solution for a dome
having
the edge r = b with a arbitrary
value e/>
and hence a
as
that in fig. 24a which abuts
against a ring having the compressive
force a t b. To avoid a local disturbance, its cross section must have
the
will be quite heavy.

Fig. 24. Domes of constant strength with skylight
choose </>0 > O. Among these solutions is the one which we obtained
for the closed shell. I f we choose </>0 so that
tan</>o
greater
values of </>0 we come to shapes as indicated by fig. 24d.
2.4 Loads without Axial Symmetry
2.4.1 General Equations
We shall now drop the assumption that all loads and stress result
ants are independent of the coordinate
(). The equations (6a-c) have
already been established for the general case. Since one of them,
eq. (6c), contains
other two . Making use of
the
obtain
r 2 ~ s m < / > +
2
lead to
a second
differential equation for N ~ . We shall come back to this
on p. 78 and we shall see then that important conclusions may be drawn
from this equation. But for the present purpose it is simpler to use the
system (27).
49
Po, Pr are arbitrary functions of cf>, 0,
they may always be represented in the form
00 00
00 00
(28)
o 1
where Pq,n . . . Prn are functions of cf> only. The first of the two sums in
every line represents
= 0 and the second
sents the antimetric part.
To find the solution of the differential equations (27) which corre
sponds to such a load, we pick out one of the terms, say
Pq, =
sin
the
where Nq,n,
How to find them
Then the general solution
for a load which is symmetric with respect to the meridian will be
00
N 8 = ~ N o n c o s n ' O ,
N q , o = ~ N q , o n s i n n O , (31)
o 1
and antimetric
and
I f we do so,
we
then
drop a common factor cosn 0 or sinn 0 from each equation. The
result is the following
2.4.2 Spherical Shell
2.4.2.1 General Solution
in
the next
Chapter with different methods, the sphere ii'l the simplest of all shells
of revolution.
I t
infinite
form.
SHELLS OF REVOLUTION
They are a good object for a study of the typical phenomena, which
we shall meet again in other shells but then only as the results of
lengthy numerical computations.
of
eqs.
(32)
,
Pr1l. •
Their sum and their difference are two independent equations for the
sum and the difference of the stress resultants,
(33)
namely:
""f' sm, , sm (34a, b)
: +
(2cotcf> - s:t/» V = 2a (pon - P ~ n - n -;::;st/> Prn).
Both are linear differential equations
of
known
l
dcf».
we
tion:
U =
sin
2
cf>
cot
n
~ dcf>].
(35a, b)
These two formulas are the general solution for the sphere, which we
shall now discuss.
Distributed Load
As example of the application of the general solution (35) we
treat a shell dome subject to a wind load. I t has become more or less
customary
P+ =
Po
p.44. - MARTIN, W. T.,
Cambridge,

This
is certainly a somewhat rough approximation, but at least it has
the advantage of acknowledging the existence of a large suction on the
lee side of the building. Therefore, the pressure pin eq. (36) should be
assumed as one half of the normal pressure given by the building codes
for a surface t right angles
to the
grations, we find
[AI +
1
as
half
sin () :
N,; = : : ~ ~ [
s
/»].
The two constants AI' BI may be determined from the condition that
the stress
nator sinS</>
has a zero of the third order, the brackets must also
have
one. The bracket for N,; will be zero, i f we put Al = - i p a. From N,; 6
comes the same result.
vanish with
any
choice of the constants. V a n i s ~ g of
the
When
the
solution
and
hoop force from (6c), we have the complete sdution of
our
problem:
N,; = -3 (1
- coscf» . ()
N,;6 = -
3
(1 +
coscf»
No = - 3
in
N ~ 6 appearing at the
springing line. The normal forces N,; have to equilibrate the moment
of the wind loads with respect to the diameter (j
= ±n/2of the spring
ing line. I f the shell happens to be a hemisphere, this
moment
the
The shearing forces N ~ 6 at the springing line resist the horizontal
resultant of the wind forces in so far as it is not resisted by
the hori
zontal components of the meridional forces. They are tangential to
the edge of the shell and therefore greatest at those places where the
.--r---=: - - -
~ - r - - - - - - - - -
Fig. 25. Spherical dome; stress resultants for wind load
()
other words, the shell carries the
loads from those zones where they are applied (near () = 0 and () = n)
to the sides. This may be recognized very clearly in a picture of the
9=-90' stress trajectories which in fig. 26
are shown in stereographic projec
tion for a hemisphere. All trajec
tories which meet the windward
meridian () = 0 have compressive
~ H - f - - H - + + t t f f l C Q H t t t l f - i + I + + - H 9 = 1 8 0
' forces; the others have tensile for
Wind ces. On the wind side, near the
6=90 '
since N = 0 there . The loads
which are applied there are carried
away by the vaultl ike compressive
Fig. 26. Spherical dome; stress trajectories
for wind load
shell. Thus most of the wind pres
sure is brought to the springing zone lying on both sides between
(J = ±nj4 and () = ±3nj4. The same thing happens to the suction in
the lee except that there the tension trajectories do the job as though
they were funicular curves. Those loads which are applied
in the
vicinity of the vertex are first carried by the trajectories with great
curvature
but
soon
are
other group, which
finally bring them down to the sides of the shell.
The so-called wind load, which we have used here, may be subject
to criticism from an aerodynamic point of view. formula (36)

53
countries, which recognize only a pressure on the windward side
and
ignore the suction. For a hemisphere the pressure distribution should
preferably have an axial symmetry to
the horizontal diameter parallel
to the wind. Such a load distribution, as might be measured in a wind
tunnel, may always be represented in the form
00
To
determine the functions Prn (cf», we only have to subject the values
on different parallel circles cf> = const to a harmonic analysis and then
collect the numerical values of the n-th Fourier coefficient
on
different
parallel circles as a tabular representation of the function Prn(cf».
Introducing it
and N ~ o n ' and if we have made
the
computation for as many values n as are necessary for convergence,
the series (31) give the stress
resultants N ~ , No, N ~ e .
2.4.2.3 Edge Load
Let us now consider the homogeneous solution, i. e., that part
of (35a; b) which remains when we put ~ = Pe = Pr =.: O.
I t describes
the stress resultants in a shell to which loads are applied only a the
edges or, perhaps, at the cf> =
0 and cf> = n.
particular solutions of the inhomogeneous equations to given boundary
conditions
them.
then apply it to cases where the distributed
load has axial symmetry but the boundary conditions have not.
I f in (35) we drop all terms containing ~ n ' Pen, Prn and determineN0
from (6c), we have
n
2
arbitrary
and
Bn.
We see at once that the case n = 1 must be treated separately,
because there
both solutions are infinite at both poles cf> = 0 and
cf> = n, whereas for n 2:
2 the
to keep the
and the
B solution
To find out what
out by an
adjacent parallel circle
on tho "",,11
wo
I and N4>B
r-
o
resultant
I
I
to
edge:
2"
o
With
N4>
sinO
we now go to
to the right)
P = n a
in
into
P = n 2 ~ (B
couples
spherical shell M = MI =
the same formulas (40)
and
find .the same force P, but in the opposite direction, and a couple
M = M2 = ~ a
55
The two forces form a couple too, and we see that the condition of
overall equilibrium
is always fulfilled, whatever the magnitudes of Al and B
1
Al
= 0, only a force P is applied at <P = 0, and
i f we choose Al
=
,
there is only a couple, but no choice is possible where there is nothing
at all.
The higher harmonics, n > 2, in eqs. (38) have singularities of a
different
type.
group of forces having infinite magnitude and canceling each other.
We shall not treat these "multi
poles" here in detail, since they do
not seem to be of practical interest
in the theory of shells.
I f the shell is not a complete
sphere but ends on a parallel circle
<P = <Po' we
to a combined normal and shear
loading of the edge, the ratio of the
two parts being fixed by <Po and n.
I t maybe used to find the stress
resultants in such cases as the one
illustrated by fig. 29.
the circumference it is a compression
and on the remainder a tension, and
the intensities of both have been
I
/'
load at a hemispherical shell
so balanced that the external forces are in equilibrium. Such forces
will occur if the shell rests on four supports of the angular width 2iX
and has to carry the edge load P between the supports, which, of
course, must be
tangential to the sphere.
We develop the edge forces, consisting of the load P and the reaction
- P ( n - 4iX)/4iX, in a Fourier series. Because of the
symmetry
forces, there will appear only the terms with n = 4,
8, 12, . . . ;
2P sinn", ()
with An and
n
: cosn (),
and
this must
that,
different from fig. 29, this edge is not at
4> = 90 0, but at some arbitrary angle 4> = 4>0' this will be accomplished
i f we choose
I t - IX
N .. = -N(J = _ 2P Si.
()
..
tangents the series converges better
the farther away we go from
the
order n
its
influence
is
felt. I t also me

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2_Stresses in Shells